Gibert's proof using the Kiepert hyperbola
Lester's circle theorem follows from a more general result by B. Gibert (2000); namely, that every circle whose diameter is a chord of the Kiepert hyperbola of the triangle and is perpendicular to its Euler line passes through the Fermat points. 
Dao's lemma on the rectangular hyperbola
In 2014, Dao Thanh Oai showed that Gibert's result follows from a property of rectangular hyperbolas. Namely, let and lie on one branch of a rectangular hyperbola , and and be the two points on , symmetrical about its center (antipodal points), where the tangents at are parallel to the line ,
Let and two points on the hyperbola the tangents at which intersect at a point on the line . If the line intersects at , and the perpendicular bisector of intersects the hyperbola at and , then the six points lie on a circle.
To get Lester's theorem from this result, take as the Kiepert hyperbola of the triangle, take to be its Fermat points, be the inner and outer Vecten points, be the orthocenter and the centroid of the triangle.
A conjectured generalization of the Lester theorem was published in Encyclopedia of Triangle Centers as follows: Let be a point on the Neuberg cubic. Let be the reflection of in line , and define and cyclically. It is known that the lines , , are concurrent. Let be the point of concurrency. Then the following 4 points lie on a circle: , , , .  When , it is well-known that , the conjecture becomes Lester theorem.
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- Dao Thanh Oai (2014), A Simple Proof of Gibert’s Generalization of the Lester Circle Theorem Forum Geometricorum, volume 14, pages 201–202. MR 3208157
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